Problem: The lifespans of seals in a particular zoo are normally distributed. The average seal lives $14.6$ years; the standard deviation is $2.1$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a seal living between $8.3$ and $20.9$ years.
$14.6$ $12.5$ $16.7$ $10.4$ $18.8$ $8.3$ $20.9$ $99.7\%$ We know the lifespans are normally distributed with an average lifespan of $14.6$ years. We know the standard deviation is $2.1$ years, so one standard deviation below the mean is $12.5$ years and one standard deviation above the mean is $16.7$ years. Two standard deviations below the mean is $10.4$ years and two standard deviations above the mean is $18.8$ years. Three standard deviations below the mean is $8.3$ years and three standard deviations above the mean is $20.9$ years. We are interested in the probability of a seal living between $8.3$ and $20.9$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the seals will have lifespans within 3 standard deviations of the average lifespan. The probability of a particular seal living between $8.3$ and $20.9$ years is ${99.7\%}$.